A Unifying Action Principle for Classical Mechanical Systems

Abstract

The modern theory of classical mechanics, developed by Lagrange, Hamilton and Noether, attempts to cast all of classical motion in the form of an optimization problem, based on an energy functional called the classical action. The most important advantage of this formalism is the ability to manifestly incorporate and exploit symmetries and conservation laws. This reformulation succeeded for unconstrained and holonomic systems that at most obey position equality constraints. Non-holonomic systems, which obey velocity dependent constraints or position inequality constraints, are abundant in nature and of central relevance for science, engineering and industry. All attempts so far to solve non-holonomic dynamics as a classical action optimization problem have failed. Here we utilize the classical limit of a quantum field theory action principle to construct a novel classical action for non-holonomic systems. We therefore put to rest the 190 year old question of whether classical mechanics is variational, answering in the affirmative. We illustrate and validate our approach by solving three canonical model problems by direct numerical optimization of our new action. The formalism developed in this work significantly extends the reach of action principles to a large class of relevant mechanical systems, opening new avenues for their analysis and control both analytically and numerically.